Question

Experiment shows that the rate of formation of carbon tetrachloride from chloroform. \(\mathrm{CHCl}_{3}(g)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{CCl}_{4}(g)+\mathrm{HCl}(g)\) is first order in \(\mathrm{CHCl}_{3}, \frac{1}{2}\) order in \(\mathrm{Cl}_{2},\) and \(\frac{3}{2}\) order overall. Show that the following mechanism is consistent with the rate law: (1) \(\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{Cl}(g)\) [fast] (2) \(\mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g)\) [slow] (3) \(\mathrm{CCl}_{3}(g)+\mathrm{Cl}(g) \longrightarrow \mathrm{CCl}_{4}(g) \quad[\) fast \(]\)

Short answer

The rate-determining step is consistent with the given rate law, and the derived rate law matches the experimental observation.

Step-by-step solution

Identify the Rate-Determining Step

The rate-determining step (slow step) is the step that determines the overall reaction rate. Here, the slow step is Step 2: \(\mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g) \[slow\]\).

Write the Rate Law for the Slow Step

Since Step 2 is the rate-determining step, the rate law will be based on this step. The rate law for the slow step is \(rate = k_2 [\mathrm{Cl}][\mathrm{CHCl}_3]\).

Express Intermediate Concentration for Cl

To express the concentration of the intermediate species \(\mathrm{Cl}\), we use the pre-equilibrium assumption from Step 1: \(\mathrm{Cl}_2(g) \rightleftharpoons 2 \mathrm{Cl}(g) \[fast\]\). The equilibrium constant expression is: \[K_1 = \frac{[\mathrm{Cl}]^2}{[\mathrm{Cl}_2]}\]. Therefore, \([\mathrm{Cl}] = \sqrt{K_1 [\mathrm{Cl}_2]}\).

Substitute Intermediate Concentration into the Rate Law

Substitute \([\mathrm{Cl}] = \sqrt{K_1 [\mathrm{Cl}_2]}\) into the rate law from Step 2: \([\mathrm{Cl}][\mathrm{CHCl}_3] = k_2 \sqrt{K_1 [\mathrm{Cl}_2]} [\mathrm{CHCl}_3]\). This simplifies to \(rate = k' [\mathrm{CHCl}_3][\mathrm{Cl}_2]^{1/2}\), where \(k' = k_2 \sqrt{K_1}\).

Verify Consistency with Given Rate Law

The given rate law is \(rate = k[\mathrm{CHCl}_3]^{1}[\mathrm{Cl}_2]^{1/2}\), which matches the derived rate law: \(rate = k' [\mathrm{CHCl}_3][\mathrm{Cl}_2]^{1/2}\). This confirms that the mechanism is consistent with the experimental rate law.

Key concepts

rate-determining step

The rate-determining step is the slowest step in a reaction mechanism. It acts as a bottleneck that controls the pace of the overall reaction. In our mechanism, the rate-determining step is Step 2: \( \mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g) \[ slow \] \). Since this step is slow compared to the others, it significantly impacts the overall reaction rate.
Understanding which step is rate-determining helps in deriving the rate law for the reaction. Always look for the slow step to determine the rate law.

rate law

A rate law expresses the relationship between the concentration of reactants and the rate of the reaction. For the slow step (rate-determining step), the rate law is based on the concentrations of the reactants involved. For Step 2, the rate law is: \( rate = k_2 [\mathrm{Cl}][ \mathrm{CHCl}_{3}] \).
The exponents in the rate law indicate the order with respect to each reactant. Here, the reaction is first-order in \( \mathrm{CHCl}_{3} \) and first-order in \( \mathrm{Cl} \). The overall reaction order is \({1/2 + 1 = 1.5}\rightarrow3/2\text{, as given}.\).
Knowing the rate law helps predict how changes in reactant concentrations affect the reaction rate.

pre-equilibrium assumption

The pre-equilibrium assumption states that the steps preceding the rate-determining step reach equilibrium quickly. In our mechanism, Step 1 \( \mathrm{Cl}_2(g) \rightleftharpoons 2 \mathrm{Cl}(g) [fast] \) is a fast equilibrium step.
This means that the concentration of the intermediate \( \mathrm{Cl} \) can be expressed in terms of the reactants using the equilibrium constant. The equilibrium expression is \( \[ K_1 = \frac{[\mathrm{Cl}]^2}{[\mathrm{Cl}_2]} \] \), which allows us to solve for \( [\mathrm{Cl}] \).
Using this assumption simplifies the derivation of the rate law by allowing substitution of intermediate concentrations with expressions involving reactant concentrations.

intermediate concentration

Intermediates are species that are formed and consumed within the reaction mechanism but do not appear in the overall balanced equation. In our reaction, \( \mathrm{Cl} \) and \( \mathrm{CCl}_{3} \) are intermediates. To derive the rate law, we need to express their concentrations in terms of reactants.
For \( \mathrm{Cl} \), the concentration is found using the equilibrium expression from Step 1: \[ K_1 = \frac{[\mathrm{Cl}]^2}{[\mathrm{Cl}_2]} \], giving \([\mathrm{Cl}] = \sqrt{K_1 [\mathrm{Cl}_2]}\). We substitute \([\mathrm{Cl}]\) in the rate law for the rate-determining step to get a final rate law in terms of \( \mathrm{CHCl}_{3} \) and \( \mathrm{Cl}_2 \).
This method ensures the derived rate law is consistent with the observed rate law.